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Optimal Rational Approximation Number Sets: Application to Nonlinear Dynamics in Particle Accelerators

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Abstract

We construct optimal multivariate vectors of rational approximation numbers with common denominator and whose coordinate decimal expansion string of digits coincides with the decimal expansion digital string of a given sequence of mutually irrational numbers as far as possible. We investigate several numerical examples and we present an application in Nuclear Physics related to the beam stability problem of particle beams in high-energy hadron colliders.

In Honor of Constantin Carathéodory

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References

  1. Baker, G.A. Jr.: The existence and convergence of subsequences of Padé approximants. J. Math. Anal. Appl. 43 (2), 498–528 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  2. Baker, G.A. Jr.: Essentials of Padé Approximants. Academic, New York (1975)

    MATH  Google Scholar 

  3. Baker, G.A. Jr., Graves-Morris, P.R.: Convergence of the rows of the Padé table. J. Math. Anal. Appl. 57, 323–339 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  4. Baker, G.A. Jr., Graves-Morris, P.R.: The convergence of sequences of Padé approximants. J. Math. Anal. Appl. 87 (2), 382–394 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bazzani, A., Todesco, E., Turchetti, G., Servizi, G.: A normal form approach to the theory of nonlinear Betatronic motion. CERN, Yellow Reports 94-02, European Organization for Nuclear Research, Geneva (1994)

    Google Scholar 

  6. Beardon, A.G.: On the location of poles of Padé approximants. J. Math. Anal. Appl. 21, 469–474 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bernstein, L.: The Jacobi-Perron Algorithm: Its Theory and Application. Lectures Notes in Mathematics, vol. 207. Springer, Berlin/Heidelberg/New York (1971)

    Google Scholar 

  8. Boreux, J., Carletti, T., Skokos, C., Vittot, M.: Hamiltonian control used to improve the beam stability in particle accelerator models. Commun. Nonlinear Sci. Numer. Simul. 17, 1725–1738 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. Borwein, P.B.: The usual behaviour of rational approximants. Can. Math. Bull. 26 (3), 317–323 (1983)

    Article  MATH  Google Scholar 

  10. Bountis, T.C., Tompaidis, S.: Strong and weak instabilities in a 4D mapping model of FODO cell dynamics. In: Turchetti, G., Scandale, W. (eds.) Future Problems in Nonlinear Particle Accelerators, pp. 112–127. World Scientific, Singapore (1991)

    Google Scholar 

  11. Brezinski, C.: Accélération de la convergence en analyse numérique. Cours de D.E.A.. Publications du Laboratoire de Calcul de l’Université des Sciences et Techniques de Lille, Lille (1973)

    MATH  Google Scholar 

  12. Brezinski, C.: Rational approximation to formal power series. J. Approx. Theory 25, 279–317 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  13. Brezinski, C.: Padé-type Approximation and General Orthogonal Polynomials. Birkhäuser, Basel (1980)

    Book  MATH  Google Scholar 

  14. Brezinski, C.: Outlines of Padé approximation. Computational Aspects of Complex Analysis, pp. 1–50. Reidel Publishing Company, Dordrecht (1983)

    Google Scholar 

  15. Brezinski, C.: Padé approximants: old and new. Jahrbruck Uberblicke Mathematik 37–63 (1983)

    Google Scholar 

  16. Buslaev, V.I., Gončar, A.A., Suetin, S.P.: On convergence of subsequences of the nth row of the Padé table. Math. Sbornik 120, 162 (1983)

    Google Scholar 

  17. Corrigan, R., Huson, F., Month, M.: Physics of High Energy Accelerators. A.I.P. Conference Proceedings, vol. 87. American Institute of Physics, New York (1982)

    Google Scholar 

  18. Daras, N.J.: Padé and Padé-type approximation for 2π-periodic L p functions. Acta Appl. Math. 62, 245–343 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  19. Daras, N.J.: On the best choice for the generating polynomial of a Padé-type approximation. BIT Numer. Math. 44, 245–257 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  20. De Bruin, M.G.: Some classes of Padé tables whose upper halves are normal. Nieuw Archief voor Wiskunde 25, 148–160 (1977)

    MathSciNet  MATH  Google Scholar 

  21. De Montessus de Ballore, R.: Sur les fractions continues algébriques. Bull. Soc. Math. Fr. 30, 28–36 (1902)

    MATH  Google Scholar 

  22. Edrei, A.: The Padé table of functions having a finite number of essential singularities. Pac. J. Math. 56, 429–453 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  23. Edrei, A.: The Padé tables of entire functions. J. Approx. Theory 28, 54–82 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  24. Eiermann, M.: On the convergence of Padé-type approximants to analytic functions. J. Comput. Appl. Math. 10, 219–227 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  25. Gilewitz, J.: Approximants de Padé. Lectures Notes in Mathematics. Springer, Heidelberg (1983)

    Google Scholar 

  26. Greene, J.M.: A method for determining a stochastic transition. J. Math. Phys. 20, 1183–1201 (1979)

    Article  Google Scholar 

  27. Jones, W.B., Thron, W.J.: Continued Fractions: Analytic Theory and Applications, vol. 11. Addison Wesley, Reading, MA (1980)

    MATH  Google Scholar 

  28. Khinchin, A.Y.: Continued Fractions. University of Chicago Press, Chicago, IL (1964)

    MATH  Google Scholar 

  29. Kim, S., Ostlund, S.: Simultaneous rational approximations in the study of dynamical systems. Phys. Rev. A 34 (4), 3426–3434 (1981)

    Article  MathSciNet  Google Scholar 

  30. Lubinsky, D.S.: Rogers-Ramanujan and the Baker-Gammel-Wills conjecture. Ann. Math. 157, 847–889 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  31. Magnus, A.: Rate of convergence of sequences of Padé-type approximants and pole detection in the complex plane. In: de Bruin, M.G., Van Rossum, H. (eds) Padé Approximation. Lecture Notes in Mathematics, vol. 888, pp. 300–308. Springer, Heidelberg (1981)

    Google Scholar 

  32. Nuttall, J.: Convergence of Padé approximants of meromorphic functions. J. Math. Anal. Appl. 31, 147–153 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  33. Perron, O.: Die Lehre von den Kettenbrüchen. Chelsea, New York (1957)

    MATH  Google Scholar 

  34. Petalas, Y.G., Antonopoulos, C.G., Bountis, T.C., Vrahatis, M.N.: Evolutionary methods for the approximation of the stability domain and frequency optimization of conservative maps. Int. J. Bifurcation Chaos 18 (8), 2249–2264 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  35. Petalas, Y.G., Parsopoulos, K.E., Vrahatis, M.N.: Stochastic optimization for detecting periodic orbits of nonlinear mappings. Nonlinear Phenom. Complex Syst. 11 (2), 285–291 (2008)

    MathSciNet  Google Scholar 

  36. Petalas, Y.G., Antonopoulos, C.G., Bountis, T.C., Vrahatis, M.N.: Detecting resonances in conservative maps using evolutionary algorithms. Phys. Lett. A 373 (3), 334–341 (2009)

    Article  MATH  Google Scholar 

  37. Polymilis, C., Servizi, G., Skokos, C., Turchetti, G., Vrahatis, M.N.: Topological degree theory and local analysis of area preserving maps. Chaos 13 (1), 94–104 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  38. Pommerenke, C.: Padé approximants and convergence in capacity. J. Math. Anal. Appl. 41, 775–780 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  39. Rosenzweig, J.B.: Fundamentals of Beam Physics. Oxford University Press, New York (2003)

    Book  Google Scholar 

  40. Schweiger, F.: The Metrical Theory of Jacobi-Perron Algorithm. Lectures Notes in Mathematics, vol. 334. Springer, Berlin/Heidelberg/New York (1973)

    Google Scholar 

  41. Shanks, D.: Non linear transformations of divergent and slowly convergent series. J. Math. Phys. 34, 1–42 (1955)

    Article  MathSciNet  MATH  Google Scholar 

  42. Todesco, E.: The use of mappings for stability problems in beam dynamics. In: Benest, D., Froeschle, C. (eds.) Analysis and Modeling of Discrete Dynamical Systems with Applications to Dynamical Astronomy, pp. 301–314. Gordon and Breach, London (1997)

    Google Scholar 

  43. Turchetti, G., Scandale, W.: Future Problems in Nonlinear Particle Accelerators. World Scientific, Singapore (1991)

    Google Scholar 

  44. Vrahatis, M.N.: An efficient method for locating and computing periodic orbits of nonlinear mappings. J. Comput. Phys. 119, 105–119 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  45. Vrahatis, M.N., Servizi, G., Turchetti, G., Bountis, T.C.: A procedure to compute the fixed points and visualize the orbits of a 2D map. CERN SL/93-06 (AP), European Organization for Nuclear Research, Geneva, Switzerland (1993)

    Google Scholar 

  46. Vrahatis, M.N., Bountis, T.C., Kollmann, M.: Periodic orbits and invariant surfaces of 4-D nonlinear mappings. Int. J. Bifurcation Chaos 6 (8), 1425–1437 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  47. Vrahatis, M.N., Isliker, H., Bountis, T.C.: Structure and breakdown of invariant tori in a 4-D mapping model of accelerator dynamics. Int. J. Bifurcation Chaos 7 (12), 2707–2722 (1997)

    Article  MATH  Google Scholar 

  48. Wallin, H.: On the convergence theory of Padé approximants. In: Linear Operators and Approximation. Proceedings of the Conference Oberwolfach (1971). International Series Numerical Mathematics, vol. 20, pp. 461–469. Birkhaüser, Basel (1972)

    Google Scholar 

  49. Weisstein, E.W.: Rational approximation. MathWorld-A Wolfram Web Resource. Available on-line at: http://mathworld.wolfram.com/RationalApproximation.html

  50. Wynn, P.: L’ \(\varepsilon\)-algorithmo e la tavola di Padé. Rendiconti di Matematica di Roma 20, 403 (1961)

    MathSciNet  MATH  Google Scholar 

  51. Zinn-Justin, J.: Convergence of Padé approximants in the general case. In: Visconti, A. (ed.) Colloquium on Advanced Computing Methods in Theoretical Physics, pp. 88–102. C.N.R.S., Marseille (1971)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Hellenic Army Academy, GR-16673, Vari Attikis, Greece

    Nicholas J. Daras

  2. Department of Mathematics, University of Patras, GR-26110, Patras, Greece

    Michael N. Vrahatis

Authors
  1. Nicholas J. Daras
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  2. Michael N. Vrahatis
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Correspondence to Nicholas J. Daras .

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Editors and Affiliations

  1. Industrial and Systems Engineering, University of Florida, Gainesville, Florida, USA

    Panos M. Pardalos

  2. Department of Mathematics, National Technical Unversity of Athens Department of Mathematics, Athens, Greece

    Themistocles M. Rassias

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Daras, N.J., Vrahatis, M.N. (2016). Optimal Rational Approximation Number Sets: Application to Nonlinear Dynamics in Particle Accelerators. In: Pardalos, P., Rassias, T. (eds) Contributions in Mathematics and Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-31317-7_6

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